Formal representation of an autoregressive model.

An autoregressive (AR) model posits a latent level whose value at each step is a noisy linear combination of previous steps:

level[t+1] = (sum(coefficients * levels[t:t-order:-1]) + Normal(0., level_scale))

sts_autoregressive(
  observed_time_series = NULL,
  order,
  coefficients_prior = NULL,
  level_scale_prior = NULL,
  initial_state_prior = NULL,
  coefficient_constraining_bijector = NULL,
  name = NULL
)

Arguments

observed_time_series	optional float tensor of shape batch_shape + [T, 1] (omitting the trailing unit dimension is also supported when T > 1), specifying an observed time series. Any priors not explicitly set will be given default values according to the scale of the observed time series (or batch of time series). May optionally be an instance of sts_masked_time_series, which includes a mask tensor to specify timesteps with missing observations. Default value: NULL.
order	scalar positive integer specifying the number of past timesteps to regress on.
coefficients_prior	optional Distribution instance specifying a prior on the coefficients parameter. If NULL, a default standard normal (tfd_multivariate_normal_diag(scale_diag = tf$ones(list(order)))) prior is used. Default value: NULL.
level_scale_prior	optional Distribution instance specifying a prior on the level_scale parameter. If NULL, a heuristic default prior is constructed based on the provided observed_time_series. Default value: NULL.
initial_state_prior	optional Distribution instance specifying a prior on the initial state, corresponding to the values of the process at a set of size order of imagined timesteps before the initial step. If NULL, a heuristic default prior is constructed based on the provided observed_time_series. Default value: NULL.
coefficient_constraining_bijector	optional Bijector instance representing a constraining mapping for the autoregressive coefficients. For example, tfb_tanh() constrains the coefficients to lie in (-1, 1), while tfb_softplus() constrains them to be positive, and tfb_identity() implies no constraint. If NULL, the default behavior constrains the coefficients to lie in (-1, 1) using a tanh bijector. Default value: NULL.
name	the name of this model component. Default value: 'Autoregressive'.

Value

an instance of StructuralTimeSeries.

Details

The latent state is levels[t:t-order:-1]. We observe a noisy realization of the current level: f[t] = level[t] + Normal(0., observation_noise_scale) at each timestep.

If coefficients=[1.], the AR process is a simple random walk, equivalent to a LocalLevel model. However, a random walk's variance increases with time, while many AR processes (in particular, any first-order process with abs(coefficient) < 1) are stationary, i.e., they maintain a constant variance over time. This makes AR processes useful models of uncertainty.

See also

For usage examples see sts_fit_with_hmc(), sts_forecast(), sts_decompose_by_component().

Other sts: sts_additive_state_space_model(), sts_autoregressive_state_space_model(), sts_constrained_seasonal_state_space_model(), sts_dynamic_linear_regression_state_space_model(), sts_dynamic_linear_regression(), sts_linear_regression(), sts_local_level_state_space_model(), sts_local_level(), sts_local_linear_trend_state_space_model(), sts_local_linear_trend(), sts_seasonal_state_space_model(), sts_seasonal(), sts_semi_local_linear_trend_state_space_model(), sts_semi_local_linear_trend(), sts_smooth_seasonal_state_space_model(), sts_smooth_seasonal(), sts_sparse_linear_regression(), sts_sum()